"One-way ANOVA" refers to "One-way Analysis of Variance". "Tukey's HSD" refers to "Tukey's Honest Significant Difference".
For Group 1="1,2,3" and Group 2="4,5,6", F=13.5, p=0.021312. As p<0.05, we reject the null hypothesis that there is no significant difference between groups.
For Group 1="1,2,3" and Group 2="4,5,6" and Group 3="4,5,7", F=7, p=0.027. As p<0.05, we reject the null hypothesis that there is no significant difference between groups. But when we look at difference within groups (1 vs 2, 2 vs 3, 1 vs 3), only Group 1 and Group 3 are significantly different by Tukey's HSD.