'He' and 'I' are both subject pronouns, i.e. they can both be only used in the nominative case and as the subject of a verb. Look at the example:

Now, look at these sentences:

In the last two sentences, we have a combined subject (or coordinated subject) which consists of two subject pronouns joined by the coordinating conjunction 'and'.

'Him and I' is grammatically incorrect because 'him' is an object pronoun and it can only be used in the accusative case. So, you cannot use 'him' in the subject position in a sentence. The correct phrase to use when 'him' is the object of the sentence is 'him and me', as 'me' is the object pronoun corresponding to the subject pronoun 'I'.