Here is the full sentence I have come across.
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てしんぜよう means "I'll do you a favor", and it is a stereotypical phrase associated with samurais. Generally, it gives an arrogant impression (or, at least it signals that the speaker genuinely thinks of themselves to be "above" the conversation partner). It is only used when the speaker is supposed to be a samurai or similar.
In this particular case, it's used to create a certain character (together with the unorthodox phrasing like "あやし" and "ぽ").
Some sites claim it can be used in daily life, but IMHO unless you are above 70 and want to come across as very arrogant, it would feel bizarre if someone uses it in a daily life context.