Plotting the results of GLM in R

I have this data plotted as a scatter plot in Excel

I had done a regression in Excel, and the p value was 2.14E-05 while the R- value was 0.32. I was told the R value was too low compared to the significance of the p value, and was told to control for the dispersion of the data by running it through R with GLM with quasipoisson error.

This gave me

glm(formula = encno ~ temp, family = quasipoisson(link = log), data = encnotemp)

Deviance Residuals: Min 1Q Median 3Q Max
-6.008 -2.431 -1.021 1.353 9.441

Coefficients: Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.005807 0.174628 11.486 < 2e-16 ***

temp 0.029065 0.006528 4.453 1.53e-05 ***

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasipoisson family taken to be 10.19898)

Null deviance: 1807.4 on 171 degrees of freedom
Residual deviance: 1620.1 on 170 degrees of freedom AIC: NA

Number of Fisher Scoring iterations: 5

How do I analyse this output?

The problem is that the scatterplot data is too dispersed, and I would like to make a scatterplot from the quasipoisson GLM output that shows less dispersed (more fitted) data points. Will this be possible?

Thank you!
Maybe Im doing this very wrong but I log transformed the response variable data (encounter number) and plotted it against the untransformed predictor variable (temperature) and got this.

I changed all the exact 0s in the response data to 0.001.

What do you mean by plots of model diagnostic plots?
Also, Im not entire sure what you are asking. Do you want to plot the fitted values?
Awesome thank you for the code!
Yes I want to fit the fitted values from the quasipoisson glm output, if that is even possible.
I was told that the quasipoisson glm will help with the outliers, and help explore the non linear relationship which is apparently shown in the scatterplot posted in the first question.