When we say someone's English sounds flat or robotic, what we mean is that their English makes everyone word sound the same! There is no change in volume, length, speed, or pitch.
We're going to focus on that last word, pitch. Never heard of it? A quick example: think of a piano. Now, a piano only has a limited number of notes. But, if you go from right to left, the sound changes! On the left, the notes are deeper. On the right, the notes are lighter and brighter.
When you speak, you can do a similar movement. Try pronouncing one side, starting deep and low. Then, keep repeating it, making it higher and higher.
What I often find with students is that their normal sentences have a range of about 3 pitches: a high pitch for stressed words, a middle pitch, and an occasional low pitch. If you listen to native speakers though, you'll find that in American English, even a "monotone" native speaker uses 4, 5, 6, or more pitches.
As a warmup exercise, try doing a progression of 5 sounds, going from low to high. Do that scale 3 or 4 times a couple times a day.
What you may notice: in general, it is easier for my students to go higher in pitch than to go lower in pitch. If that is the case for you too, don't worry! In fact, it is helpful to know what the issue may be, and now you know what to target with future exercises.
There are also many common pitch patterns in English that are worth practicing and applying to your own American English speech consistently! Here is a video that will provide with you examples of exercises to do to start getting a more natural American English sound when you talk: