Japanese i-adjectives (速い, 強い...) are inherently predicates, so in theory they don't need another verb "to be" to support. Here, the extra ある is a kind of rhetorical reflection of the will to be (in a state), or consciously keep being.
It can be used in the same way with na-adjectives and nouns in the form ～である. Of course, である can be also just an expanded form of ～だ, so the form is less exclusive than i-adjectives (whose conjugations, like 速かった is contraction of 速くあった too, but it never expands anymore).
強くあろうとする strive to be strong/tough
生涯気高くあった remained dignified throughout one's life
80歳になっても健康でありたい want to stay healthy at the age of 80
良き教育者であれ Be a good educator!
This is a relatively literary, solemn expression. Also, the difference with ～くなる/～になる is that なる implies you are currently not in the state (80歳になる means you aren't now), which is not the case with ～くある/～である (健康でありたい doesn't mean you aren't now).
[adjective]くある also appear when you attach a focusing particle to an i-adjective.
= 確かに薄いが、… / 薄いことは薄いが、…
It's slim to be sure, but too heavy to carry around.