chicken: 2 legs
goat: 4 legs
total: 58 legs

guess & check:
20+5 (=25)
20 chickens & 5 goats
= 20x2 + 5x4
= 40 + 20
= 60 (wrong!)

guess & check:
21+4 (=25)
21 chickens & 4 goats
= 21x2 + 4x4
= 42 + 16
= 58 (right!)

(a) 42 legs

(b) 4 goats

Assume that a chicken has 2 legs and a goat has 4 legs.

There cannot have been 25 chickens on the farm, because there would only be 25 x 2 = 50 legs altogether.

(Hmmm...the question never mentioned that there cannot be other animals...)

There cannot have been 25 goats, because there would be 25 x 4 = 100 legs and there are only 58 legs altogether.

Let's do some guessing.

24 chickens, 1 goat
24 x 2 + 1 x 4 = 48 + 4 = 52 (FAIL)

23 chickens, 2 goats
23 x 2 + 2 x 4 = 46 + 8 = 54 (FAIL)

22 chickens, 3 goats
22 x 2 + 3 x 4 = 44 + 12 = 56 (FAIL)

21 chickens, 4 goats
21 x 2 + 4 x 4 = 42 + 16 = 58 (YES)

20 chickens, 5 goats
20 x 2 + 5 x 4 = 40 + 20 = 60 (FAIL)

Other combinations will never get you 58 legs (for every chicken that is replaced by one goat, two legs are added to the total count).

Hence, there are 21 chickens and 4 goats.

The chickens contribute to 42 of the 58 legs.