It is Spiders (2000). See here on IMDB.
The trailer even shows a character in a suit like within the picture above, in agony. It gives the impression that the spiders breed and use the humans as hosts.
Stars include Lana Parrilla, currently the Mayor/Evil Queen on Once Upon a Time.
Here's the IMDB plot summary:
A DNA experiment on a rare breed of spider is taking place on a NASA space shuttle, when a freak meteor shower engulfs the shuttle, causing everything to go horribly wrong. One survivor is found on the ship and watched over in a secret location deep in the California desert. The problem continues, as the survivor isn't alone, as another deadly spiders climbs out of him and goes on a rampage around the ship. Curious reporter Marci Eyre must now survive, escape and warn everyone before the spider reaches outside the desert.
Here's the youtube link of the trailer in French