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They have tiny pieces that are considered to be a choking hazard. There have been some high profile and tragic deaths among children due to choking on toys with small pieces. So the United States is especially stringent in regulating children's toys. The Kinder Surprise Eggs are considered especially dangerous as they are embedded in a food product. So a young child might think the toy is edible as well. Wikipedia.

Note that the manufacturer has worked around this ban by introducing the Kinder Joy

It has plastic egg-shaped packaging that splits into two, one half contains layers of cocoa and milk cream and the other half contains a toy.

Because the edible and non-edible parts are separated, this can and is sold in the US.

And since someone suggested in a comment protectionism might have been at play... That's unlikely in this case because the US law enacting this generic prohibition goes back to 1938; it was part of the Food, Drug, and Cosmetic Act enacted in the wake of the "Elixir" sulfanilamide scandal.
The Kinder Surprise was introduced in the late 1960s.

The law as summarized in a paper on this topic, authored by an FDA employee:

The regulations in the Federal Food, Drug, and Cosmetic
Act on the hazards of nonnutritive objects found in confectionary
products have been in effect since 1938. Specifically,
Section 342(d) (1) of the Act states, “If it is confectionery, and
has partially or completely imbedded therein any nonnutritive
object: Provided, That this clause shall not apply in the case of
any nonnutritive object if, in the judgment of the Secretary as
provided by regulations, such object is of practical functional
value to the confectionery product and would not render the
product injurious or hazardous to health.”

On the other hand, Nestle did lobby to have the US law changed in the 1990s when they introduced their "clone" Magic Ball product.

Nestle also attempted a political lobbying blitz to change the law, including trying to get Congress to embed wording in an Agricultural Appropriations bill that would have made their product legal, with Representative George Nethercutt of Washington state, whose district just so happened to include a large Nestle plant employing hundreds of voters, championing the tweak in wording.

They did not prevail though.

And since another answer brought up choking hazard: that is a somewhat different issue. Despite FDA's opposition, the Magic Balls were actually sold in the US for approximately two months (in 1997) because Nestle's product managed to pass the explicit choking tube test (16 CFR § 1501) of the U.S. Consumer Product Safety Commission. Mars (a competitor of Nestle) engaged however in counter lobbying through Carol Tucker Foreman, an former Assistant Secretary of Agriculture in the Carter Administration who argued that the Nestle product barely passed the CPSC choking test. Following her arguments numerous state attorney generals issued warnings against the Magic Balls, and the lobbying effort that would have explicitly allowed it floundered.

The mingling prohibition in the 1938 law is not absolute:

The Federal Food, Drug and Cosmetic Act of 1938 prohibited the marketing of any candy if it had imbedded in it a ''non-nutritive object'' unless the object had a purpose. The F.D.A. has interpreted that exception to apply to items like the sticks in lollypops. Boxes of Cracker Jack, which have long had small toys next to the snack, are governed by a different regulation because the toy is not imbedded in the food product, but is ''commingled.''

Nestle argued that the FDA's interpretation of the law was too strict. When this failed to convince the FDA directly, Nestle tried to have Nethercutt's bill amendment force the FDA to change its interpretation. (As I already mentioned, this effort failed too.)

So the ban on Kinder Surprise is combination of law and its interpretation by the FDA. The agency actually has a fairly long import ban list of similar candy products it considers illegal under the 1938 law. The preamble to the actual list does say it was prompted by the Kinder Surprise (and some unnamed "similar articles") brought to their attention in this respect.

Also, in the 1950s and 60s the FDA was battling vending machines that were mixing candy with "trinkets":

A seizure was instituted against a lot of gum and candy intermingled with trinkets. It was alleged that the mingling of trinkets with candy resulted in adulteration in violation of Section 402(a)(1) of the Federal Food, Drug, and Cosmetic Act which provides that food shall be deemed adulterated "if it bears or contains any poisonous or deleterious substance which may render it injurious to health". The seizure was contested. The U.S. District Court upheld the seizure, but on July 24, 1951 the 4th Circuit Court of Appeals reversed this decision, holding that the trinkets were not contained within the gum or candy. (Cavalier Vending Corporation v. United States, 190 F. 2nd 386 (4th Cir. 1951) (reversing United States v. A Quantity of Candy Containing Trinkets, et al., 95 F. Supp. 490 (E.D. Va 1951).)"

In 1964, HEW and FDA initiated legislation intended to clearly outlaw the practice of vending confectionery mingled with trinkets through an amendment to Section 402(d) of the Act. Section 402(d)(1) was amended to provide that confectionery shall be deemed to be adulterated if it has partially or completely imbedded therein any non-nutritive object.

Considering the amendment, both the House and Senate Reports mentioned the Cavalier case, and both mentioned that FDA had asked for legislation to put a stop to the intermingling of trinkets with candy unless the trinkets were wrapped in cellophane or otherwise separated from the candy. Both the House and Senate said that such legislation should not be allowed, because the vending machine operators had one of the lowest liability insurance rates in the food industry, indicating a low incidence of injuries.

[...]

because of the decision of appellate court in Cavalier Vending Corp. v. U.S., and the legislative history of P.L. 89-477, amending Section 402(d)(1), the Administration is not in a position to take regulatory action because of the intermingling of trinkets with confectionery, so long as the trinkets are not completely or partially imbedded in candy or gum.

This was also well before the Kinder Surprise was even commercialized (1968, if Wikipedia is correct on that.) So the FDA hard line on "trinkets" wasn't something specifically targeting Kinder Surprise. Also the FDA accepted "commingling" (at least in the case of vending machines) after losing some court cases as noted above.

Apologies for the poor quality, but this photo of a poster from the FDA History Office gets the point across.

The text reads:

During the Great Depression, candy with a "prize" inside -- usually a coin or lead trinket -- was very popular. However, they could be deadly when swallowed by small children. These photos taken by Dr. Chevalier Jackson, a pioneering laryngologist, helped convince Congress to approve a provision in the 1938 Food, Drug and Cosmetic Act and still in force today, which prohibits the sale of trinkets embedded in confections.

Even though a different answer was already accepted, I believe it might be wrong. In a video by Youtube channel "Today I found out" it's explained that this isn't so much because of the choking hazard, but because of a law that forbids non-edible elements in food as long as they don't serve a practical purpose (e.g. the stick in lollipops serves to hold it, so it's allowed, but the egg containing the present in a kinder surprise doesn't have a purpose which is why the eggs are illegal).
It looks like the video was already made some time ago, so I don't know if the answer has been disproved since, but here's the link: https://www.youtube.com/watch?v=ffzbfO0c5Qs