A kindly-looking old man stood up and extended his hand for a handshake.

Literal

Kind-looking old-person [topic-は] stood-up-and, handshake [object-を] sought [toward me-きた].

~そうな ('looking like') makes 優しい into a visual impression rather than a known fact. 求めてきた uses ~てくる to show the action directed toward the speaker. The scene unfolds in two steps: standing up, then reaching out.