Do cats and dogs actually understand the words we say?

Yes—but usually not in the same way people understand language.
Cats and dogs can learn that particular sounds predict particular things. “Walk,” “dinner,” or “outside” may become meaningful cues because they’re repeatedly followed by an event. They may also use your tone, gestures, where you’re standing, and the time of day. So a pet reacting to “outside” might recognize the word, notice you reaching for the leash, or both.
Some dogs can learn the names of many objects, and research suggests cats can recognize familiar words, including their own names. But that’s different from understanding sentences and grammar as humans do. Most pets are responding to learned associations and context—not following every word in a conversation.
A simple way to check is to say the cue in a neutral voice, without reaching for the related object or moving toward the door. You can also try it in a different room or at an unusual time. If your pet still responds, the word itself is probably part of what they’ve learned.