What powers a Nest thermostat?
The Nest runs on power supplied by your furnace. Typically the red wire is +24v. It doesn't need battery backup partly because its scheduling and other configuration data are stored online. It may also have some non-volatile memory for basic data.

In some cases, the common (C) wire isn't present. In your case, you have an extra conductor that should serve that purpose if properly connected. Here's more on that.

No need to call in the pros. If you find that something doesn't work, contact Nest. If all else fails, hook the old thermostat back up until you figure out the issue.