Translation:There is one dog at his house.
When counting small animals, all of the numbers end in -hiki except the number 3 (which ends in -biki) and the numbers 1, 6, 8, and 10 (which end in -piki)
It's the same for all animals. It changes depending on what is easier to say - so it depends on the sound of the number preceding the counter. Some sounds go together easier than others so the pronunciation of counters for animals changes accordingly. It's quite logical really.
Without the は we would just be saying that he has one dog in his house/household, it would just be a general sort of statement with no particular significance. With the は we're making it clear that we're talking about how/focusing on what things are like in his house eg. In HIS house, there's one dog - other people might have more dogs, no dogs, dogs AND a cat, no animals at all etc, but as for HIS house there's one dog. The は marks "かれのいえに" as the particular subject matter that we're focusing on. Hopefully I've explained it well?