Will adding this switch make my Internet connection faster?

Not simply because a switch has been added. It gives you more wired connections; your Internet speed is still constrained by the broadband service, router, connected devices and the links between them.

Replacing an old 100Mbps switch with a Gigabit model can remove a local 100Mbps bottleneck, which may make a real difference to file transfers or a faster Internet service. The switch can remove a restriction; it cannot manufacture bandwidth upstream.