For one, it can be used for caching of filesystem, making it much faster to launch applications or handling of multiple files. Secondly, it obviously allows you to have more stuff running concurrently. Thirdly, as screen resolutions keep growing so do the RAM requirements for images and video, so the more RAM you have the higher-res content that can be handled efficiently. Even just a single RGB 8bit-per-channel 1080p image, when uncompressed in memory, would take ~8.3 megabytes if it includes an alpha channel or ~6.2 megabytes without such. Grabbing a sequence of images, doing any sort of image manipulation on them or whatever, and you’re suddenly easily looking at a few hundred megabytes of RAM-usage.
How does more RAM translate into battery-savings, then? Well, for one, caching allows the system to retain more data instantly-accessible reducing the needs for the system to access mass storage. Also, since the data is more readily accessible the faster whatever a task the CPU is given can be completed — if the data is cached the data only has to go from RAM to CPU cache, whereas if it isn’t cache the data has to go from mass storage to RAM to CPU cache, an obviously longer route. Also, the more RAM there is available the more data that can be kept uncompressed, meaning that the less time the CPU has to keep juggling between compressed and uncompressed data and the more time it can idle.