Math and realism to the rescue!
The least populous state in the US is Wyoming, in the ballpark of 570,000 residents according to last year's census. In this least populous state, a petition with 25,000 signatures represents about 4.4% of the population. Wyoming is not on the list of states with a petition to file for secession, ergo every other state with a petition has a higher population, ergo the percentage of signatures to total population is even less (not even accounting for the fact that most of the petitions haven't even reached 25,000). These petitions do not, by -any- stretch of imagination, represent the will of any kind of significant portion of the population in a single state with one of these petitions. It's just a lot of hot gas, and as the media is inclined to do, simply bringing it up because it's a 'newsworthy story'.
Not to mention the petitions themselves have no legal bearing and are non-binding. The White house could simply choose not to even -respond- to any of them, let alone actually do anything about them. And even then, there are very few states that could realistically actually seceed, in the economic sense. Most of them are completely bound to the economy of the country as a whole. And -then- that's not getting into how the seceeded state would loose every piece of profession military it has; as we saw after the Civil War, state military was phased out in favor of centralized national military. Which, if we're delving this far into absurd-hypothetical land, means that the US could actually say "Okay, you can seceed. Now you're a completely independant country with no allies and no treaties with anyone." And then turn around and re-invade the state to occupy it.
This entire thing is just a silly farce by some whackjobs who are sad their dude didn't win, and as already noted, simply don't get what democracy means. This isn't even worthy of an eyebrow-raise to me. Give it a few weeks, and people will forget this ever happened. (now, if a Congressperson of one of the states were to start throwing support behind one of these petitions, -then- I'd raise an eyebrow and take notice.)