Literally, a "shutter" is a protective covering of a window. So, figuratively, having the shutters up means protecting something or oneself.

这个是英式吧

A shutter is like a screen of hinged panels on a window that are opened or closed. Open shutters make it posiible to look through the window, and for light to go through. Closed shutters block light, and you can't see through In this image, on the right the shutters are opened by pulling down the rope. On the left, the shutters are closed. Here, in this context, the shutters are a metaphor. The man feels like there is an obstacle/screen between him and the woman emotionally. The man is not showing his thoughts and feelings to the woman. Note: In english, whether a person means up is to closed as down is to open is dependant on context but usually up => closed, down => open. For example, 'windows up' means windows are closed, that is, there is an obstacle in the way. 'Wall down' means the wall is now broken, there is no longer any obstacle. However, in this audio, as the man is implying an obstacle between him and the woman emotionally, 'shutters down' is usually more appropriate. It is a mistake on the behalf of the speaker. Spoken word language often does not follow all rigid or flexible rules of a particular language, so understanding usually comes with exposure and practice.