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For that example I would be thinking 60 x 25 = 1500 (4 x 25=100, so that's the simple part for me) and then 3 x 25 = 75, and add the two products.

In general I find mental arithmetic like this easiest if you start from the most significant digits. So you just need to keep track of a running sum and which multiplications you still need to add.

Here's the calculation as I would do it in my head, more or less:

25 * 63 = (20 + 5) * (60 * 3)

= (20 * 60) + (5 * 60) + (3 * 20) + (5 * 3)

= 1200 + (5 * 60) + (3 * 20) + (5 * 3)

= 1500 + (3 * 20) + (5 * 3)

= 1560 + (5 * 3)

= 1575

If division can be involved, why not:

= ((25 * 4) * 63) /4

= 100 * 63 /4

= 6300 / 4

= (6400 - 100) / 4

= 1600 - 25

= 1575