English does this too, and you have never noticed: two slices of bread, three sheets of paper, a pack of cards. You cannot say 'two breads'. Bread is a substance, and to count it you need a word that carves it into countable units.
Thai treats almost every noun the way English treats bread. หมา is not 'a dog' so much as 'dog' in general, and to count it you need the word that says what shape of thing you are counting. For animals that word is ตัว.
- หมาสองตัวmǎː sɔ̌ːŋ tuːa
two dogs
noun, number, classifier, in that order
- คนสองคนkʰon sɔ̌ːŋ kʰon
two people
some nouns are their own classifier
How to learn them without drowning
There are hundreds and you need roughly twenty. Learn the classifier as part of the noun, the way you would learn gender in French. Never learn หมา alone; learn หมา with its classifier ตัว.
And when you are stuck, อัน is the general-purpose fallback for small objects. Using it where a more specific classifier belongs sounds slightly childish, but it is always understood, and it is far better than freezing mid-sentence.