人気があがっていった。
They became popular. / They have been popular. (from maggie)
*@Jon As the site the chart is from explains, "ていく" expresses, among other things, "progression or change over time from a given temporal reference point onwards" ("基準時から基準時以後への推移・変化を表わす場合には「Vていきます」を使います").
Apply this meaning of "ていく", "人気があがっていった。" more or less translates to "The popularity grew (and grew)."
Since the "would" in "They would become popular." indicates "future-in-the-past" or "prospective" aspect (I think), it doesn't match the meaning of "ていく". (refering to this translatation:They would become popular)