My understanding of this is that the contigs are used as a backbone for assembly. Using these scaffolds/contigs as a backbone, the assembly becomes more of a mapping - the raw (trimmed) reads stretch the contigs, and that's how it helps with gap closure. The same thing could happen with the mapping + re-assembly, but this would require many more steps because no graph is constructed, whereas in SPAdes it is.
In numbers - this also helped me to better understand - say we have a 1000 bp long contig. A 250-bp long read overlaps the terminal 100 bp of the contig. This means that there is a 150 bp long stretch over the tip of the 1000 bp contig, and on this stretch, other reads can overlap - something that doesn't happen when you map.