(This uses the notation used by Noether in her Lagrange's theorem writeup)
Define the (left) cosets of H in G as all the distinct sets gH, where g is an element of G. Note that any 2 cosets have the same number of elements: multiplication (on the left) by gh-1 performs this 1-1 mapping from hH to gH. We claim that the cosets partition G.
It is clear that every element of G is in some coset. Indeed, since the identity element e is in H, if g is in G then it is also in gH, which is a coset.
To prove the theorem, all that remains is to show that for every g,g' in G, either gH=g'H or the 2 sets are disjoint. So suppose the intersection of the sets is non-empty, and contains some element k. Then k=g h = g' h' for some h,h' in H, so g'-1g = h' h-1 is in H. But this means that every element g' j of g'H (j in H) may be expressed as g g-1g' j, and since g-1g' j is in H, this shows g' j is in gH too. Switching rôles of g and g' proves equality.