start counting intersections from any corner and you will notice that they increase/decrease in raw 1:3:5:7:9:7:5:3:1. Sum them up to get precisely 41 intersections for 4x4 square. For square 5x5 not shown here and outside of this question's picture (imaginery) this will be 1:3:5:7:9:11:9:7:5:3:1, by summing up we get 61

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Is there a faster way to do it? I didnt really understand the answer.

I categorize the intersections as one of two types:

(1) those created by diagonal lines
(2) those created by horizontal/vertical lines

You can see that each identical square has 1 intersection created by diagonal lines.
In this type of grid, which contains identical squares and has 'N' rows and 'N' columns, there are N^2 intersections from diagonal lines

Now look at the top row. Notice that there are 5 vertical lines which intersect the top-most horizontal line.
In this type of grid, which contains identical squares and has 'N' rows and 'N' columns, there are (N+1)^2 intersections created by horizontal/vertical lines.

In a 5x5 grid, N=5. There are N^2 = 25 intersection points from diagonal lines

There are also (N+1)^2 = 36 intersection points from horizontal/vertical lines.

There are no other types of intersections, so the total # of intersections = 25+36=61.