This has to do with intron splicing. Remember GTAG. This mutation induced an AG closer where it was supposed to be, so some of that intron just became an exon.
As described in the question stem, this mutation occurs within an intron (a gene segment which is transcribed [DNA->RNA] but not translated). RNA splicing enzyme(s) grab RNA and “loop it”; an intron is cut out and the exons on either side of the intron are adjoined, like this:
exon1—intron—exon2 => exon1—exon2
Typically, this splicing occurs at the very edges of the intron (what I denoted with the “—” character). But in our case, a mutation within the intron is causing RNA splicing enzyme to recognize a new site: the splicer cuts within the intron (instead of at the very edge, as it should). So, we get something that looks like this:
That’s a totally different mRNA molecule, and it's going to make our β-globin protein look (and behave) awfully strange.