Construct an AND gate from XOR gates
The XOR operation is bilinear: inverting either of the inputs always inverts the output.

This property is preserved even if you cascade multiple XOR gates (and, optionally, constant inputs and/or NOT gates) together: inverting any input always inverts the output.  Of course, you can construct more complex circuits where a single input can be split and fed into multiple gates, but all such circuits can be expanded into an equivalent tree of XOR gates, possibly with the same input appearing multiple times.  For each input, one of the following two cases will then hold:


if the input value appears an odd number of times in the XOR tree, inverting that input inverts the output;
if the input value appears an even number of times in the XOR tree, its effects on the output cancel out, and the output ends up being completely independent of that input.


In any case, the only kinds of functions you can construct by combining XOR gates are those where each input either never affects on the output, or will always invert the output when the input is inverted.

Now consider an AND gate: if the first input is 1, the output is equal to the second input, whereas if the first input is 0, the output will always be 0 regardless of the second input.  This is impossible to implement with XOR gates: in any XOR-only circuit, toggling the second input should either always change the output or never change it.

All this can be expressed concisely in mathematical terms: XOR is a multilinear operator, and composition of multilinear operators is always multilinear.  AND is not a multilinear operator, so it cannot be obtained by combining XOR operators.