Clarification on proof: Order of left cosets equal
Because this is the definition of having the same cardinality.
First, you should know that two sets $A,B$ have the same size (by definition) if there is a bijection $f:A\to B$ (in this case there is also a bijection $g:B\to A$).
You can understand why this is the definition in the case that $A,B$ are finite by using a drawing.
Second, $|aH|=|H|=|bH|$ (since the size of every coset is equal to the size of $H$) and thus all cosets have the same cardinality.