Q:

Why do we make assumptions in the T test?

I am now studying Student's T-test and trying to understand the assumptions of it. 
I haven't seen any sources that describe why such assumptions are made. 
For a T-test with two samples, why are both groups assumed to have the same variance?

A:

Ultimately the reason you make assumptions for hypothesis tests is to be able to compute the distribution of the test statistic when the null hypothesis is true. This then allows the calculation of either critical values or p-values.
Often some assumptions may appear reasonable because of a priori considerations (it often seems clear from the nature of the situation that the responses should be independent or nearly independent, for example). 
Some assumptions are less clear; it's not always obvious that the variances should be the same. However, note that strictly speaking for hypothesis tests we only need this to be the case when the null hypothesis is true (though we may sometimes want to perform additional calculations that may further require this to be the case under the alternative, and interpretation of a rejection will often be simpler in that case).
It's quite often the case that if we're testing for say something like a treatment effect, that if there isn't one then it would be anticipated that the the distributions are identical under the null. In that case an assumption of equal variances under the null would generally be pretty uncontroversial.
There are tests that don't assume that the variances are the same under the null hypothesis (Welch-Satterthwaite t-tests for example) but we can't exactly compute the distribution of the test statistic under the null hypothesis in that case -- that test is only approximate. 
In this particular case, the equal variance version of the test allows us to actually compute the exact distribution of the statistic (given the other assumptions hold), when the null hypothesis is true -- it's a t-distribution -- and hence get p-values. 

