What does this plot of predicted versus standardised residuals mean? 
I have been taught that residuals should be randomly distributed relative to the predicted values of the DV so that there should be no pattern. Can anyone please tell me what this output means?
 A: Forgetting about standardization for a moment, the definition 
$\text{residual} = \text{observed} - \text{fitted}$ 
implies for each distinct observed value a distinct line with negative slope on this plot. If standardization implies scaling by the same positive constant, then that remains true. 
I see in your example four parallel lines, so I infer four distinct values in your response or outcome variable (*). It doesn't matter here how good or bad the model is, as that variability in model merit affects only where the fitted values are on those lines. Such a pattern is therefore inevitable in any residual versus fitted plot of this kind; it's just a question of whether you notice. 
Incidentally to your question, but likely to be important to your project: 


*

*The response variable can be decoded with a little guesswork: the name accidentslogsqt suggests a raw variable number of accidents, which is likely to be a count 0, 1, 2, 3, ... Whether it's people, cars, pets, other valuable items, the same rule applies. 

*I guess you're worried by severe skewness and a belief that the response should be normally distributed, which is a myth. 

*So, what you have done? Perhaps something like $\log\root \of {y + 1}$ with a preliminary fudge to fix the zeros. 

*The logarithm of a square root (of a positive number) is just half the logarithm of the original number, so the square root here does nothing that the logarithm does not do to skewness or heteroscedasticity or nonlinearity. 
I am open to correction on those guesses, and I will stop guessing there, but will suggest that plain regression is definitely a bad idea for such a response variable, if only because nothing stops prediction of a negative number of accidents, which is absurd. I suggest Poisson regression as tailored to a counted response (and applicable more widely, but that is a different story). 
(*) You prefer the term dependent variable (or DV) and I advise against that too: (a) Terms dependent and independent variables are heavily overloaded in statistics and mathematics generally. (b) Independent variables generally aren't independent in any meaningful sense. (c) There are better, more evocative terms. (d) DV is horrible in-crowd jargon that does not help communication. 
