Like other idiomatic sayings, "a low threshold of suspicion" doesn't make much literal sense, so what it's referring to might not be immediately obvious. However, an understanding of the individual parts of this saying sheds light on its figurative meaning.
Having a Low Threshold
Having a low threshold for something means you have a low tolerance for it, or alternatively that you are more sensitive to it and experience it more easily than the average person. To use one common example, if you have a low threshold for pain, it could mean that you cannot tolerate very much pain, or that you feel pain more readily than other people do.
Merriam-Webster defines suspicion as "suspecting something wrong without proof or on slight evidence," or "a state of mental uneasiness and uncertainty." People experiencing suspicion may be right or wrong in their feelings that something isn't right, but either way, they feel there's more to the story than they have been explicitly told or can prove.
Put the two parts of this saying together, and you understand its meaning--a person with a low threshold of suspicion is easily moved to doubt what they are told, or what they've observed and fear, or assume that something is wrong. They can experience these suspicious feelings more often and in more situations than other people.