How does a euro rail pass work in Switzerland?

It's Eurail - not Euro Rail and it's best left behind along with the traveler's cheques. Buy your point to point tix and save yourself money and time and effort.

For Switzerland if you plan on any high mountain excursions like to the Jungfraujoch then invest in a Swiss Half Fare Card which will save you 50% on all trips in Switzerland. The savings alone on the Jungfraujoch trip will practically pay for the Half Fare Card.

Not the best way at all, but that is just a personal opinion.
Eurail pass works in Switzerland, until you reach Interlaken Ost, then it only gives you a 25% discount on tickets to Lauterbrunnen, and the mountains above.

If it were me, I'd buy my ticket from Budapest to Vienna in Budapest, where they are cheap bought in the local currency. Vienna to Innsbruck I'd buy well in advance to get a discount for a nonrefundable ticket. Same for Innsbruck to Luzern. In Switzerland, I'd add up the cost of all my trips, and if the list price was over 240 CHF, I'd buy a Half Fare Card for 120 CHF. Then Basel to Paris, I'd buy advance nonrefundable discount tickets. With a Eurail pass, I'd have to buy a separate seat reservation for that train, which may or may not be available.

A Eurail Global Pass covers all those trains. Unless a special deal comes along this spring, you'd probably choose the version for 7 travel days within a month, about $377 per adult in 2nd class. You've listed 6 travel days above but could use the 7th day for a side trip from a city.

You need a seat reservation on the TGV to Paris, which most people catch at Basel (reservation fee about $33 in 2nd class, or cheaper if you make an extra connection and catch the fast train at Strasbourg). You also need a seat reservation if you choose a named Glacier Express departure for part of your route toward Zermatt, but you may also choose an unreserved regular train on the same tracks, depending on schedule preference. Looking Up Train Schedules and Routes Online gives you the Deutsche Bahn train schedule link and tips for using it. It will prioritize the fastest connections, unless you specify a detour toward a scenic route.

From Interlaken to Lauterbrunnen and further up that valley to the mountains, the Eurail pass provides 25% train and lift discounts. If you're taking a side trip during your stay in Lucerne, it can cover lower level trains (on counted pass travel days, does not cover buses) and provides 50% discounts on lake boats and higher mountain lifts. In both cases, discounts don't require using a counted travel day. See more about pass coverage in Switzerland.