CIDR allows IP addresses to be allocated to institutions more efficiently. Under the classful system, fully half the address space was reserved as 128 class A, 16.8 million-address networks. Very few institutions have need for that many addresses, and under the classful system, there was no way to break those class A networks down into smaller chunks that could be assigned to different institutions. Even the biggest Fortune 500 companies (that aren't telcos/ISPs) would only have several hundred thousand addressable devices, so millions of addresses would go to waste for each institution that had a class A.
Take, for example, Ford Motor Company (the American car maker). Ford was assigned a class A network back in the day. So Ford has 16,777,216 addresses allocated to it, even though it only has 160,000 employees worldwide. Does Ford really need over 100 publicly-routable addresses per employee? Probably not, since they're not even a tech company, much less a telco or ISP or other network provider. I could maybe believe 10 addressable devices per employee MAX, which means Ford is wasting over 15 million addresses. Unfortunately Ford was "grandfathered in" when CIDR went into effect, so they're still wasting those addresses. But thanks to CIDR, there were only a couple dozen grandfathered-in wasteful class A allocations, and most of the rest of what was previously the "class A" network space (remember, this was half of all IPv4 address space) has been doled out to Regional Internet Registries (RIRs) around the world, and broken down into smaller blocks to be assigned in APPROPRIATELY-sized chunks to institutions that needed it.
Remember that under the classful scheme, ½ of the entire IPv4 address space was reserved as 128 class A networs, ¼ was reserved for class B networks, ⅛ was reserved for class C networks, 1⁄16 was reserved for class D (multicast), and the last 1⁄16 was reserved for class E (undefined future use).
Because there was no way to break the class A half of the address space down into smaller units that could be assigned to different institutions, the numbering authorities couldn't in good conscience allocate any of the class A networks to anyone except perhaps the biggest telcos and mega-ISPs. And the reasonably-sized chunks (classes B and C) only made up ⅜ of the address space, so all of that would have been depleted very quickly.