connected to two switches
That's what we call a 3-way switch. The traditional configuration of a 3-way switch has something like:
- Incoming Power
- Switch 1
- Switch 2
The key is that between those 2 switches there are (not counting ground) 2 wires, but really 3 wires with one of them either only connected to one of the switches (hot going to 2nd switch or switched hot coming back from 2nd switch) or not connected to either switch (neutral just coming along for the ride). With typical cables using standard colors (black/red/white), that can get very confusing, because the white wire is not neutral!
The end result is that a dimmer, timer, motion detector or smart switch on a 3-way installation needs to know how to handle that extra wire. Some handle it just like ordinary switches, some require a special compatible switch to replace the 2nd switch, some use just one wire, etc.
The device you purchased is R00-RNL06-0TW, which is for a basic, single switch installation. Leviton makes a similar device, the R00-RDL06-0TW. In a quick search, it seems that Farm & Fleet, Home Depot (note: Home Depot typically has installation manuals online which makes this type of thing easy to figure out) and Leviton all have slightly different model #s available. But the common - and key - denominator is RDL06 instead of RNL06.
If you get something else (fancier Leviton or another manufacturer), make sure it specifically lists "3-way" and "LED". Whatever you get, read the instructions carefully before you get started. 3-way switches can be confusing.
And just in case you think "I'll just figure out some other way around this and avoid another trip to the hardware store", NO. When I was a kid, we had 3-way switches in the dining room and installed a timer. The timer was not designed for 3-way switches. For many, many years (probably until the house was torn down a few years ago to build a new house), we were told "don't touch the other switch, it will mess up the timer!" Of course, there was no internet to consult in those days :-(