"Mom has three older brothers."
When you count objects of a sentence in Chinese, there are always words called 'measure words' that are used. The measure words follow the actual number of the object--in this case, 3. Think of when you count the number of deciliters or cups you need when you are making a recipe, such as "10 deciliters of flour". The measure word in this case is 'deciliters', and the object is 'flour'. It is the same concept in Chinese, except it applies to Everything when you are counting. Most of the time, for 'objects' of a sentence related to people, 个 is used. Hope this helps!
I struggle to understand that as well, if some chinese speakers pass by it would be nice to have an answer :)
Here's my high school's Mandarin textbook lol: https://drive.google.com/file/d/0ByUq645I-K69aENSeTZicnE3SGs/view
I also recommend the sites archchinese.com and wordreference.com (both can be used for looking up definitions but the first one can also be used to learn how to write Chinese characters stroke by stroke).
I imagine it as any Chinese noun doesn't have the property of 'countness' in the first place like in English the word "car" means one car by itself but you want to say several cars you just put 's' at the end. Chinese doesn't care about it. I'm not a Chinese speaker though, it's just a speculation from my part.
Honestly I accept such things as it is because at the beginning, sure, it's confusing and alien but further on it will be as it is. I know it from my experience of learning English. Long time ago I didn't understand what articles were. Why are they needed? Now I think it's a useful thing and it's just wrong not to use them.
In English you can't say, for instance, "one water", "two milks" etc. You need to use a measure word in order to complement your information, thus you could say "one glass of water" or "two cartons of milk". Chinese is just like that but all nouns require those words, so you can't say "one older brother", "two older brothers" as you would do in English, you have to use a measure word to make those words countable. 个 or 個 is the measure word for units of people, hence 妈妈有三个哥哥 means "mom has three 【units of people】 older brothers."