Those points are due to cases where Fatalities = 0 so the log(Fatalities) returns -Inf. Notice in the plot below that I get similar points at x values of 2,6, and 8. You can remove these points with the filter() function from the dplyr package.
part_c <- Terrorism_Data %>% filter(ftmpop > 0) %>%
ggplot(aes(y = log(ftmpop), x = log(gdppc))) +
geom_point(color = "steelblue2") +
labs(x = "ln(GDP per Capita)", y = "ln(Fatalities (per million persons)") +
ggtitle("ln Relationship")
Example showing similar points:
library(ggplot2)
#> Warning: package 'ggplot2' was built under R version 3.5.3
DF <- data.frame(X = 1:10, Y = log(c(10,0,44,23,105,0,87, 0, 33, 6)))
ggplot(DF, aes(X, Y)) + geom_point(size = 3)

Created on 2019-10-27 by the reprex package (v0.3.0.9000)