Does a negative value of intercept suggest that the regression line provides poor fit to the data? why? and why not?
No, a negative value of intercept does not suggest that the regression line provides “poor fit to the data”. Why not? Because the intercept is not a measure of fit for a regression. It is a value that represents one element of the model, not the quality of the fit.
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. WP:EN s.v. Goodness of fit
[...] y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. WP:EN s.v. y-intercept
See KDG's answer for an illustration.
As others have said, intercept is not a measure of fit at all, so a negative intercept does not indicate a poor fit. But -- you asked if a negative intercept suggests a poor fit. In some contexts, the answer is clearly "yes". If $X$ and $Y$ are physical quantities such that $Y < 0$ makes no physical sense, and furthermore values of $X$ close to 0 are non-outliers which are part of your data set, then a linear model $Y = mX + b$ with $b < 0$ (by a significant amount) won't be a good model for your data. Note that such a model will have large residuals for those $X$ values close to 0, so the poorness of fit will show up in the standard measures.
But in other contexts, a negative intercept is unproblematic even if $Y < 0$ makes no sense. As an example, R's built-in sample dataset trees gives height, volume and girth of 31 black cherry trees. If you do a linear regression of volume as a function of girth (
lm(formula = Volume ~ Girth, data = trees)) you get a linear model with a negative intercept. The fit isn't bad (even though a quadratic model would be better) with $R^2 = 0.93$. You could use it to get reasonable predictions of the volume of such a tree in terms of its girth. In this case, the negative intercept simply indicates that it is unreasonable to extrapolate a relationship which (roughly) holds among mature trees to mere saplings. But - you don't need a negative intercept to tell you that such extrapolation is problematic. I wouldn't trust any physical model with $X$ near 0 unless the data itself involved such values.
The intercept value has no relation to goodness-of-fit.
To see if a fit-line is good fit for the data you need to calculate the distance between the fit-line and all the data points (as shown in @KDG's answer).
One metric for calculating this is Root Mean Square Error (RMSE).