His argument has five steps on the way to objectivity (why moral principles have to be ojective for Kant is a different question):
Step 1: Only rational beings have a will.
The will is thought as a capacity to determine itself to action in
conformity with the representation of certain laws. And such a capacity
can be found only in rational beings. (4:427)
Why is that so? The reasons are spread out through his work, but essentially it is about will as a causality of freedom as different from a causality of nature (the will can form itself independently from nature). And only rational beings have the capacity of spontaneity, i.e. independence from the causality of nature, as rationality is defined as exactly the faculty to be spontaneous (see CPR A445|B473 ff. and A533|B561 ff.). You could say that iff one is rational, one has a will in the sense of this definition [Wille, i.e. not Willkür].
Step 2: To determine the will, one needs an end.
Now, what serves the will as the objective ground of its self-determination is the end, and this, if
it is given by mere reason, must hold equally for all rational beings.
By contrast, what contains merely the ground of the possibility of an
action the effect of which is an end is called the means. (ibid)
Pretty self-explanatory. If you want something, the objective ground of determining the will (read: at all) is that you want to achieve an end. This is not about how the will is determined in particular (this includes means, maxims, and all this subjective stuff). The second sentence is important later on: by mere reason means only by the part of the will that is shared by all rational beings qua being rational, that's why it holds equally for all rational beings.
Step 3: Distinction between subjective and objective ends.
The subjective ground of desiring is the incentive, the objective ground of willing
the motivating ground; hence the difference between subjective ends,
which rest on incentives, and objective ones, Which depend on moti-
vating grounds that hold for every rational being. Practical principles
are formal if they abstract from all subjective ends; they are material
if they have these, and hence certain incentives, at their foundation. (ibid)
This is an important distinction if we are speaking about objectivity of morals: If I have an incentive to perform an action, there is no reason to think that every rational being would necessarily share it. That's why we need a formal principle that can nevertheless serve as a motivating ground.
Step 4: If the end of an action is to achieve a certain, determined effect, it is only relative to that effect and the subjective interest in it, i.e. a subjective end in need of an incentive.
The ends that a rational being intends at its discretion as effects of its
actions (material ends) are one and all only relative; for merely their
relation to a particular kind of desiderative faculty of the subject gives
them their worth, which can therefore furnish no universal principles that are valid as well as necessary for all rational beings, or for all willing, i.e. practical laws. That is why all these relative ends are the
ground of hypothetical imperatives only. (4:427-8)
Long story short: If my willing is dependent on willing the outcome, my willing will only be so if I have an incentive to achieve the desired effect. This means it cannot serve as objective (formal) principle, but is necessarily material and therefore subjective.
Step 5* (if you do not read carefully): Rational beings are our best guess for objective ends and if there is such a thing, it's them.
[...] rational beings are called persons, because their nature already
marks them out as ends in themselves, i.e. as something that may not be
used merely as a means, and hence to that extent limits all choice (and
is an object of respect). These are therefore not merely subjective ends,
the existence of which, as the effect of our action, has a worth for us; but
rather objective ends, i.e. entities whose existence in itself is an end, an
end such that no other end can be put in its place, for which they would
do service merely as means, because without it nothing whatsoever of
absolute worth could be found; but if all worth were conditional, and
hence contingent, then for reason no supreme practical principle could
be found at all.
This one is a bit tricky. He does not really present an argument why we should be ends in ourselves.
Step 5 (Kant explained): Because we realise that we and all other rational beings are setting ends and have lawgiving will, we see that we all - by virtue of setting ends and being able to do so rationally - are ends in ourselves.
That [being an end in itself] is how a
human being by necessity represents his own existence; to that extent
it is thus a subjective principle of human actions. But every other rational being also represents its existence in this way, as a consequence
of just the same rational ground that also holds for me;* thus it is at
the same time an objective principle from which, as a supreme practical
ground, it must be possible to derive all laws of the will.
*Here I put this proposition forward as a postulate. The grounds for it will be
found in the ﬁnal section. (4:429)
I cannot quote and explain the deduction here, but the idea is that he argues that we realise that our capability of rationality and acting under the idea of freedom makes us a part of an intelligible world that we share with all rational beings and that it is this part that enables us to have free, autonomous will just like any other rational being as well.