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It's going to rain starting tomorrow night, so I'll take a spare umbrella with me.
Literal
Tomorrow [possessive-の] night [from-から] rain [because-だから], spare-umbrella [object-を] do go.
置き傘 is a compound noun — 置く (to place, leave) + 傘 (umbrella) — meaning an umbrella you keep stashed somewhere like at work or school for sudden showers. 置き傘をしていく is a slightly unusual construction: it uses する on the noun to mean 'make it so that one has a placed umbrella,' with the ~ていく auxiliary suggesting the action carries outward toward the destination. The volitional ~いこう ('let's go / I'll go') expresses the speaker's resolution.