[I read this question as wanting: "The quantity x^x = y looks like it should always decrease as x decreases from 1 to 0, but actually has a minimum. How can one think about this expression to see that this will be the case?"]

Thinking of exponentials as growth rates may help. This first section will be a general introduction to this mentality.

First, imagine some system that will change over time. A typical example is an experiment with bacteria reproducing in a petri dish for some amount of time. The number of bacteria in this system can be expressed in an exponential form as A = A₀ * b^p, where A₀ is the initial number of bacteria present; b, the base, is the "growth factor", meaning the amount of change that should occur after a single time step; and p, the power (aka the index), is the "duration", meaning the number of time steps that the system will be allowed to change.

As a specific example, consider A₀ * 2³. With the growth-and-time mentality, this means that the number of bacteria will double (base = 2) after a single time step, but we will let it keep growing for 3 time steps (power = 3). So at the end of the experiment we will have 8 times (2³ = 2*2*2 = 8) as many bacteria as at the beginning. This thought process can be extended to fractional values.

A fractional exponent means that we stop the experiment after a portion of a time step: A₀ * 2^0.8 means it will double after a full time step, but the experiment is stopped after 0.8 = 80% of a time step). A base less than 1 means that the system "grows smaller", or decays with time, rather than getting larger.

Now go to the problem of x^x (which I'll write as 1*x^x), where the values of x that we care about (per the question) are those between 0 and 1. For x close to 1, such as x = 0.9, this becomes 1 * 0.9 ^0.9 .

In the growth-time point of view, this can be interpreted as decaying for nearly a full time step. But 0.9 is not that different from 1, so even after decaying for (nearly) a full time step the amount of stuff present does not decrease by much. At the other end of the range, take x = 0.0001 to give 1 * 0.0001 ^0.0001. Since the base is small, this is like saying that the system decays quickly with time, but because the power (amount of time) is also very small, the system doesn't decay for very long.

Because the decay does not occur for much time, the final value will not be decreased from the initial by much. Now pick a value of x near the middle of the range, such as x = 0.5. Then the expression is 1 * 0.5 ^0.5. In this case, the system decays at a significant rate AND is allowed to decay for a significant amount of time. Thus, one can intuitively see that the ending value will be noticeably smaller than the initial value. Put all of this together by considering the final values from these three cases: at both ends of range (x = 0.9 and x = 0.001), the final value is "large", while in the middle of the range the final amount will be "small". Thus, starting at x ~= 1 and decreasing to x ~= 0, the quantity x^x will initially get smaller before increasing when x is small enough. This is because at the ends of the range x^x doesn't decay for very long (x ~= 0 case) or does decay very quickly (x ~= 1 case), while in the middle it both decays rapidly and for a longish time.

Here are a couple of sites which try to explain exponents in an intuitive way. This Feynman lecture may also help with thinking about exponents and how their rules are built. This page has a little on the math showing the coordinates of the minimum.