Swansea are 20th and after the sacking of Paul Clement earlier this week.
"If the situation we're in isn't a wake-up call, then I don't know what will be," said Britton, 35.
"The manager's gone and if people are sleeping, they need to wake up right now because we have a big battle on our hands. I need to get a bit of confidence back in the team, some clarity, just make things clear for the players.
"One of the reasons I agreed to take this role is I believe I can have an effect on the team in the immediate future."