There are a lot of thorough answers already provided on promoting business that you may want to search.
Instead of repeating the all the tips, this answer hopefully addresses your specific question.
In your profile, you indicated that you go to dog parks, other parks, and play on lawns inside your community.
I'd suggest the easiest way to talk about your services is to start a conversation about their dog. Compliment them on their dog (it might be appearance - grooming, leash, harness, clothes, water bowl or energy level or obedience, etc) Ask about the dog, such as breed, age, name, the story how they found each other, etc. If the dogs are enjoying each other's company or you think it would be a good match, let them know if they ever need a pet sitter at their home or yours, you offer that service. The other person can then ask about your experience or for your card. Basically, you're just introducing yourself and dog to them and theirs, sharing your love for dogs, and letting them know you're available. After some time, it should feel really natural.