What does logistic model assumption "The true conditional probabilities are a logistic function of the independent variables" mean? Here a list of model assumptions for logistic regression is given.
Number 1 is:

The true conditional probabilities are a logistic function of the independent variables.

What does this technically mean?
 A: Recall that one of the assumptions of linear regression is linearity, which is expressed as 
$$\mu_{y|x_1, \dots, x_k} = \beta_0 + \beta_1x_1 + \dots + \beta_kx_k$$
The logistic regression assumption that you just stated may be interpreted in a similar way.
A: There are a variety of ways that $P(Y=1|X_1=x_1, X_2=x_2, ..., X_p=x_p)$ may operate.
Let's simplify by considering a single independent variable (single predictor), $x$. Then as $x$ increases, how does $P(Y=1)$ change?
Even if we restricted ourselves to monotonic nondecreasing functions there are many possible choices (essentially any cdf might be used).
The logistic corresponds to saying $P(Y=1) = [1+e^{-(\alpha+\beta x)}]^{-1}$ (though this can be written in several other forms), a family of relationships indexed by two parameters which change the position of the place where $P(Y=1) = \frac12$ and how steep the function is there (the scale).

The function in blue is the logistic. The function in green is a normal cdf, producing a "probit" rather than "logit" model; somewhat similar in shape but it turns more sharply near the top and bottom of the range. The remaining curve is a Weibull cdf. On the logit scale all the logistic functions would be linear but the other possibilities would not be linear on that scale.
To respond to the request in comments -- aside from the legend, something like the above plot can be produced by:
 x  <- (0:100)/10
 xx <- (x-5)/.8
 y  <- 1/(1+exp(-xx))
 plot(x,y,type="l",col=4,ylab="P(Y=1|X=x)")
 lines(x,pnorm((x-5)),col=3)
 lines(x,1-exp(-(x/5)^1.4),col=2)
 abline(h=c(0,1),col=8)

