Deadheading removes the plant's signal to stop producing flowers, so yeah, it genuinely does work for most roses. The key is cutting just below the spent bloom, right above a leaf node (where leaves branch out from the stem). You want to cut at a 45-degree angle, about a quarter-inch above that node. For climbing roses specifically, you're actually in a tricky spot because they tend to flower once per season on old wood, so deadheading won't trigger the same repeat-blooming cycle you'd get from a hybrid tea or floribunda rose. That said, removing dead flowers still prevents the plant from wasting energy on seed production, and you might get sporadic secondary blooms later in the season.
The difference between rose types matters more than people realize. Repeat bloomers (like hybrid teas or many shrub roses) will keep pushing out new flower clusters if you deadhead consistently through the growing season. Climbing roses and old-garden varieties often have just one big flush, so deadheading helps but won't magically turn them into constant bloomers. For your climbing rose, I'd suggest deadheading the spent flowers now, but don't expect a June-level rebloom in August or September. What you might get instead is cleaner-looking growth and maybe a few bonus flowers in late summer if you're lucky. Just make sure you're not cutting too far down the stem - stay close to that leaf node, otherwise you'll leave dead stubs that look messy and can invite disease.