Why is Least Recently Used better than FIFO in relation to page files?
If you mean in terms of offloading memory pages to disk - if your process is frequently accessing a page, you really don't want it to be paged to disk, even if it was the very first one you accessed. On the other hand, if you haven't accessed a memory page for several days, it's unlikely that you'll do so in the near future.
If that's not what you mean, please edit your question to give more details.
Treat the RAM as a cache. In order to be an effective cache, it needs to keep the items most likely to be requested in memory.
LRU keeps the things that were most recently used in memory. FIFO keeps the things that were most recently added. LRU is, in general, more efficient, because there are generally memory items that are added once and never used again, and there are items that are added and used frequently. LRU is much more likely to keep the frequently-used items in memory.
According to temporal locality of reference, memory that has been accessed recently is more likely to be accessed again soon.
There is no single cache algorithm which will always do well because that requires perfect knowledge of the future. (And if you know where to get that...) The dominance of LRU in VM cache design is the result of a long history of measuring system behavior. Given real workloads, LRU works pretty well a very large fraction of the time. However, it isn't very hard to construct a reference string for which FIFO would have superior performance over LRU.
Consider a linear sweep through a large address space much larger than the available pageable real memory. LRU is based on the assumption that "what you've touched recently you're likely to touch again", but the linear sweep completely violates that assumption. This is why some operating systems allow programs to advise the kernel about their reference behavior - one example is "mark and sweep" garbage collection typified by classic LISP interpreters. (And a major driver for work on more modern GCs like "generational".)
Another example is the symbol table in a certain antique macro processor (STAGE2). The binary tree is searched from the root for every symbol, and the string evaluation is being done on a stack. It turned out that reducing the available page frames by "wiring down" the root page of the symbol tree and the bottom page of the stack made a huge improvement in the page fault rate. The cache was tiny, and it churned violently, always pushing out the two most frequently referenced pages because the cache was smaller than the inter-reference distance to those pages. So a small cache worked better, but ONLY because those two page frames stolen from the cache were used wisely.
The net of all this is that LRU is the standard answer because it's usually pretty good for real workloads on systems that aren't hideously overloaded (VM many times the real memory available), and that is supported by years of careful measurements. However, you can certainly find cases where alternative behavior will be superior. This is why measuring real systems is important.