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Blurry pic + lack of details does not help. Super imposing arrows or lines would be helpful.
Are you referring to the thin, brown shoots? I can't tell from the pic but are they covered with a fine light colored fur or "hair"? If so, then they are probably adventitious roots. These are roots that originate from a stem or node in stem plants and they help when propagating the plant. You can ignore them.
Not exactly sure what plant that is.
In general, you would trim the stem below where the roots appear. You want to ensure that the cutting has roots to help it become established wherever it is planted.