An spoken English sentence is a finite string of phonemes. The set of allowable phonemes is finite. Given a finite set X, the set of all finite strings of elements of X is countable.
(The latter statement holds because for any given n, the set X_n of all strings of length n is finite. So you can count the members of X_0, then count the members of X_1, and so on, and by continuing on in that way you'll eventually count out all members of X. You never run out of numbers to assign to the next set because at each point the set of numbers you've already assigned is finite (it's smaller in size than X_0, ..., X_n combined, for some n).
In fact, even if you allow countably infinitely many phonemes to be used, the X_n sets will still be countable, if not finite, and in that case their union is still countable: to see that, you can take enumerations of each set put them together as columns an a matrix. Even though the matrix is infinite in both dimensions, it has finite diagonals, so you can enumerate its cells by going a diagonal at a time, like this (the numbers reflect the cells' order in the numeration):
1 3 6 10 15
2 5 9 14
4 8 13
7 12
11
However if you allow sentences to be countably infinitely long, then even when you only have finitely many phonemes, the set of all sentences will be uncountable, because in that case each countably infinitely long sentence can be mapped to a real number represented as an expansion in some base, and you can apply Cantor's diagonal argument. The "just count out each X_n separately" argument doesn't work in this case because it only applies to the sentences of finite length.)
(The latter statement holds because for any given n, the set X_n of all strings of length n is finite. So you can count the members of X_0, then count the members of X_1, and so on, and by continuing on in that way you'll eventually count out all members of X. You never run out of numbers to assign to the next set because at each point the set of numbers you've already assigned is finite (it's smaller in size than X_0, ..., X_n combined, for some n).
In fact, even if you allow countably infinitely many phonemes to be used, the X_n sets will still be countable, if not finite, and in that case their union is still countable: to see that, you can take enumerations of each set put them together as columns an a matrix. Even though the matrix is infinite in both dimensions, it has finite diagonals, so you can enumerate its cells by going a diagonal at a time, like this (the numbers reflect the cells' order in the numeration):
However if you allow sentences to be countably infinitely long, then even when you only have finitely many phonemes, the set of all sentences will be uncountable, because in that case each countably infinitely long sentence can be mapped to a real number represented as an expansion in some base, and you can apply Cantor's diagonal argument. The "just count out each X_n separately" argument doesn't work in this case because it only applies to the sentences of finite length.)