I asked someone the number of ways to put $10$ distinct to $2$ identical boxes and the reply was : just take one object and put it into one box and compare. So, that's just $\frac{2^{10}}{2}$.
I didn't understand that "take one object" and "compare" portions.
On
Briefly, distinguish the boxes as box zero and box one. Now you are counting the number of ten digit binary strings ($2^{10}$) except that it doesnt matter which box is which (you can switch all the digits) so $\frac{2^{10}}{2}$
As far as what that person meant, consider asking the source to elaborate.
One way to approach the question is to pick some subset of the objects to put in one box. There are $2^{10}$ ways to do this. You then put the rest of the objects in the other box. As the boxes are identical, you have counted each configuration twice, so divide by $2$.
Another approach is to pick one object and put it in a box. That makes the boxes different. Now pick some subset of the remaining items to put with the first one, which you can do in $2^{(10-1)}$ ways. Put the rest in the other box and you are done.