Early consoles and home computers worked with extremely limited firepower. In those days, not only was the hardware less capable, there really wasn't room for expandability to make machines like the Commodore VIC-20 or Famicom more capable. Or wasn't there? Cartridge based software might have some nostalgic value for enthusiasts today (and it might also be making a comeback soon) but it also played an important role in pushing gaming machines beyond their limits. As The 8-Bit Guy and The Obsolete Geek explain, carts had a huge advantage over magnetic tape: They could house additional hardware.
Practically, more chips could lead to everything from better sound to polygon rendering for 2.5D games. Although machines of this era could only "see" a limited amount of RAM at a time, some software-side trickery also allowed carts to pack in way more than the maximum amount of data and tell the processor which chunk to look at.
It didn't always pan out — as in the Atari 800 which had a second cart slot for exactly this purpose, but a handful of carts were made to work this way, one of which was a game based on the band KISS. But when it did pan out, it paved the way for some of the most memorable games of the era. Classics like Super Mario RPG and Starfox would never have been possible without the carts themselves improving the hardware they were designed for.