Explaining Fixed and Random Effects Let's say that I am trying to predict the Sepal Length in 
Iris data from Sepal Width, Petal Width, and Petal Length variables.
Say, we noted that Petal Length and Petal Width are grouped as described by the Species variable.
I am using the following mixed-effects model for this.
libraray(lme4)

fit <- lmer(Sepal.Length ~ Sepal.Width + Petal.Length +  Petal.Width +
             (1 + Petal.Length + Petal.Width | Species), data = iris)

My question is how to explain this model concisely in plain language?
For example:
In our mixed-effect model, we considered Petal Length and Petal Width
to be random effect variables as they may contain variation that can be
explained from the Species variable. Further, we assumed all three
Sepal Width, Petal Length, and Petal Width are fixed-effect variables.
 A: What about

We assume that the sepal length of these three species of iris flowers
  depends on petal length and petal width in a somwhat similar way.
  Thus, it makes sense to learn from one species to better understand
  the relationship in the other two species.

If you want to further go into implied assumptions like exchangeability (there's of course also the linearity of the relationship, normality of random effects and so on), perhaps

We assume that we know nothing up-front about whether we would expect
  one of these three species to have a higher average sepal length, or a
  sepal length that depends more strongly on the petal length and width.

Of course, in the particular example things are especially tricky, because you only have three species so estimating the between species standard deviation is difficult. Additionally, you can know a lot from the literature without looking at the data that would make exchangeability questionable.
A: I typically suggest to people to write it is as follows:

To account for the correlations of sepal length measurements within a flower species we fitted a linear mixed-effects model. In the fixed-effects part, we included the linear effects of sepal width, petal length, and petal widths. In the random-effects part, we included intercepts and linear random slopes for petal length and petal width.

In case you have done so, you can also write:

The assumptions of the model were evaluated using residuals plots. The random-effects structure was selected using likelihood ratio tests, and p-values for the fixed effects coefficients are based on F- and t-tests using the Satterthwaite's approximation for the degrees of freedom. The models were fitted in R using packages lme4 and lmerTest.

