You can attach them to the masonry walls using an appropriate adhesive like Liquid nails, applied with a caulking gun. For added insulation, you may want to spend a few extra dollars and to buy panels of rigid foam insulation that can be glued against the walls between those furring strips. Then attach your sheetrock and paint it or put up your paneling and you’re good to go.
Since this basement is not entirely underground - it includes a walk-out entrance on one side -- new egress windows bring natural light into the basement. A simple Roman shade in fun fabric softens the space and brings all the colors of the room together. Egress windows provide an emergency exit; most building codes require one in every bedroom and in habitable basements. Adding one will likely increase the value of a home, as it provides safety, natural light, and increased airflow.
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