The only real answer is that the US Supreme Court, in interpreting the constitution, and specifically the argument that the 13th Amendment prohibits a draft for compelled military services has totally rejected that argument. For many years now the US has not used a draft, and it is obviously possible for the US to have an enduring and powerful military without any draft, which was perhaps not apparent to the Justices in 1918. A draft had been common in this country from the colonial period, through the Revolution, the Civil War, World Wars I and II, the Korean War, and the Vietnam War. In reaction to the problems during the Vietnam War, and the great opposition to any draft at that time, the US has not used a draft since, although it retains a legal requirement to register for a possible draft, and the legal authority to impose one should it be thought wise. Note that this was not because of the 13th Amendment. Note also that compelled service by the citizens (or residents) in a locality, particularly to fight fires and floods, when the usual forces are inadequate to that end, has been commonly used. Such compulsory service has never been thought to be prohibited by the 13th Amendment.
Also, as mentioned in comments, citizens can be compelled to do jury duty, which could in theory be considered "involuntary servitude" but has never been thought to be prohibited by the 13th amendment.
The answer by Trish (now deleted) thoroughly described the many differences between a slave and a drafted soldier. Still, drafted military service might be thought to be a form of involuntary servitude. But the Court (and the laws and other courts as well) have not treated it as such. In the Selective Draft Law Cases, 245 U.S. 366 (1918) the Court thought the idea that compulsory military service constituted involuntary servitude was so wrongheaded that it thought a very brief mention sufficient to refute this contention. It wrote (at 245 U. S. 390):
Finally, as we are unable to conceive upon what theory the exaction by government from the citizen of the performance of his supreme and noble duty of contributing to the defense of the rights and honor of the nation, as the result of a war declared by the great representative body of the people, can be said to be the imposition of involuntary servitude in violation of the prohibitions of the Thirteenth Amendment, we are constrained to the conclusion that the contention to that effect is refuted by its mere statement.
It is clear from the text of that opinion that the justices thought that the existence of a power to draft soldiers was essential to the implementation of the constitutional power (article I section 8):
- To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water;
- To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
- To provide and maintain a Navy;
- To make Rules for the Government and Regulation of the land and naval Forces;
The opinion discusses the history of compelled military service in the United States, in the colonies before there was a United States, and in Great Britain before that. The opinion says that:
Compelled military service is neither repugnant to a free government nor in conflict with the constitutional guaranties of individual liberty. Indeed, it may not be doubted that the very conception of a just government and its duty to the citizen includes the duty of the citizen to render military service in case of need, and the right of the government to compel it.
Further, it is said, the right to provide is not denied by calling for volunteer enlistments, but it does not and cannot include the power to exact enforced military duty by the citizen. This however but challenges the existence of all power, for a governmental power which has no sanction to it and which therefore can only be exercised provided the citizen consents to its exertion is in no substantial sense a power.
One may disagree, but that is the law of the land as interpreted by the final body authorized to make such interpretations, the Supreme Court, and it remains good law today.