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I am not sure if this is the type of direct proof that you are looking for, but here it goes. I will start with a more general theorem: Let $M$ be a CANCELABLE monoid, and $K$ be the left adjoint to the forgetful functor, $U:GROUP\rightarrow MONOID$. Then $BM$ is homotopy equivalent to $BK(M)$. The way I like to see this is to think of both monoids and (therefore groups) as categories. By $B$, I mean the nerve of the category that turns the category into a simplicial set. Now I find these sorts of nerve theorems are much easier to see in the world of simplicial sets. To see this particular theorem, it suffices to try to build a minimal fibration (that is a fibrant replacement). In the case of a monoid, all of the inner horns are filled, and we must only find out how to fill the outer horns. But these will just be adding in the inverse that are not yet in the monoid. Further the minimal fibration condition will ensure that horn fillers are unique. Essentially, what you are doing is to perform a geometric version of the $K$ functor in the category of simplicial sets. As an interesting exercise, it would be good to take the minimal fibration associated to $B\mathbb{N}$ and see that you get the simplicial set, $B\mathbb{Z}$.

Now for the James construction that you mention (even though this is not part of your question, it is worth mentioning) their is a simplicial set version of this called the Milnor FK construction. What you do is you start with a reduced simplicial set, $X$ (a simplicial set with one vertex). We then define a simplicial group called $FK(X)$ the n-th group is the free group on the elements of $X_n$ modulo the image of the iterated degeneracey, $s_0^n(pnt)$, where $pnt$ is the vertex.

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I am not sure if this is the type of direct proof that you are looking for, but here it goes. I will start with a more general theorem: Let $M$ be a monoid, and $K$ be the left adjoint to the forgetful functor, $U:GROUP\rightarrow MONOID$. Then $BM$ is homotopy equivalent to $BK(M)$. The way I like to see this is to think of both monoids and (therefore groups) as categories. By $B$, I mean the nerve of the category that turns the category into a simplicial set. Now I find these sorts of nerve theorems are much easier to see in the world of simplicial sets. To see this particular theorem, it suffices to try to build a minimal fibration (that is a fibrant replacement). In the case of a monoid, all of the inner horns are filled, and we must only find out how to fill the outer horns. But these will just be adding in the inverse that are not yet in the monoid. Further the minimal fibration condition will ensure that horn fillers are unique. Essentially, what you are doing is to perform a geometric version of the $K$ functor in the category of simplicial sets. As an interesting exercise, it would be good to take the minimal fibration associated to $B\mathbb{N}$ and see that you get the simplicial set, $B\mathbb{Z}$.

Now for the James construction that you mention (even though this is not part of your question, it is worth mentioning) their is a simplicial set version of this called the Milnor FK construction. What you do is you start with a reduced simplicial set, $X$ (a simplicial set with one vertex). We then define a simplicial group called $FK(X)$ the n-th group is the free group on the elements of $X_n$ modulo the image of the iterated degeneracey, $s_0^n(pnt)$, where $pnt$ is the vertex.
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