I get the feeling that you are looking at the log odds and not the probabilities.
Lets assume you are predicting y ~ x. If you take the coefficients and apply the equation b0 + b1 * x, the result does not give you a probability, but the log odds which needs to be converted to a probability. One easy way would be to use the predict function with type as response.
See the example below (note that the model used in the example does not make any sense).
d <- data.frame(
y = sample(c(0, 1), 30, replace = TRUE),
x = rnorm(30)
)
fit <- glm(y ~ x, family = "binomial", data = d)
summary(fit)
predict(fit, type = "response")