Inference in linear mixed model Let have an imaginary income data that comes with three categorical (Binary) variables, having TV (yes=1,no=0) at home, washing machine (yes=1, no=0) and age (old=1,young=0) as well as the income for the response, y. Assume that the data is collected mounthly and repeated for three months and the following linear mixed model (in nlme or lme4 format) is fitted
y = Tv + Wm + Age + Tv:Wm + Tv:Age + Wm:Age + (1|Months)

Let all terms be significant after fitting the model.
Then, my question is the interpretation of the terms. Actually, I have confused myself. Is Tv significant means that it is significant CONDITIONAL on the other effects or not? How about if I want to answer this question: is there any difference between the income of the people who have a WM at the young age? 
I would deeply appreciate if somebody explains the mathematical reasoning in addition to the actual inference.
 A: When you have a random effects model, the interpretation of the variable's coefficients is the same as if it was a linear regression model (let's assume you already know how to do that). 
What changes in a random effects model is that you're inserting a term in the model to account for the intraindividual variance. This new term follows a Gaussian distribution with mean = 0 and variance = $\sigma^2_{b_i}$. This means we're interested only in the variance of this term. And the likelihood is given by, where both the distributions of $y_i$ and $b_i$ are gaussians:
\begin{align*}
      L_{i}(\theta_{i} ; y_{ij}) & = \int f(y_{i} | b_{i}) \cdot f(b_{i}) {\rm d}b_{i} \\
     \end{align*} 
About the part of yout question about conditionality, is only $y$ that might be conditioned on the random effects, or:
$$ y | b \sim N(X\beta + Zb, \sigma^2 I)$$
A: I'm not going to explain the math (others here can do that much better than I can) but:

is there any difference between the income of the people who have a WM
  (and those who do not) at the young age?

is not the exact question this model answers.  It controls for age, so it holds age constant.  If you want to look just at young people, you should look at a subset of the data. 
What you have now, with the interaction term, let's you look at whether the effect of having a WM is different at different ages. 
