Just to add to what's been already said, the standard way to assess prediction capacities of a model, is to split data set in training and test subsets, fit the model to the training data, and then check its prediction on test data. 

1) "fit" means that the model coefficients are chosen so that the observed data is most likely assuming the model with these coefficient values. This does not mean that the data is far less likely assuming different coefficient values. The p-values that are reported for the coefs give the probability of "less likely" data (than the observed) assuming your model being restricted in the value the respective coefficient can take (usually 0, what in a linear model corresponds to a model without the respective term). If this probability is small, then the assumption that the coef has the assumed value (e.g. 0) does not seem adequate. This is what is meant with "significant".

2) the model does predict the repsonse, assuming the model with the estimated coefficiens. In so far: yes. The question is how accurate and how precise these predictions are. Is the model useful for the purpose?

Just want to add a point to Jochen's comment. When the purpose is to do prediction, the overall performance is much more important than individual predictors. You might end up with a model with no significant predictors (very rare), but this model must win in other model selection criteria (i.e. adjusted R^2, AIC, BIC, MSE etc)  

Evenif the model is significant with p = 0.002, R^2 indicates less than 25% variation is explained by the model and thus wont give good prediction. Secondly if individual coefs for each independent variable is not significant implies that these variables cannot predict the dependent variable. In my opinion the model cannot be chosen for studying multiple correlations nor prediction.

All the best

Just to add to what's been already said, the standard way to assess prediction capacities of a model, is to split data set in training and test subsets, fit the model to the training data, and then check its prediction on test data. 

..I would usually dismiss the R2 and such fit measures. Issue is...R2 is designed for linear methods (linear regressions). These methods usually FAIL for predictions; they are not even valid for most nature applications (see McArdle paper 1988 on that very issue). So how to compute R2s for Neural Networks or Classification & Regression Trees, or for boosting, bagging and ensembles ? R2 is just not cutting it, certainly not pseudo-R2s (=approximated and assumed R2s) . So I propose to move away from model fits, usually, and focus on inference from predictions, and thus, ROCs etc etc.

See Leo Breiman 2001  and other machine learning citations.

A more relevant discussion to have than R2: Why do R2s last that long ? Has to do with tenure and the lame administrative bodies, failing unis & their publishers, NSF, Royal Academies and their evaluations; no progress and no turn-over even. It's a cast! But why not instead hiring and tenuring people on their skill for creating best predictions and ROCs, let's say ? Western Society at a loss and on the way down, again ?

Wish us (=the earth) luck.